LDpred Models for Bayesian Score Construction

Why LDpred Uses Both GWAS Results and Linkage Disequilibrium

A marginal GWAS effect at one SNP can partly reflect effects at correlated SNPs. Treating each marginal estimate as though it were the SNP’s independent effect can therefore produce distorted score weights. LDpred addresses this by combining GWAS summary statistics with an LD matrix and a Bayesian prior on the underlying effects.

The central idea is to estimate each SNP’s effect using information from the full set of correlated summary statistics. The prior represents a belief about how many SNPs have nonzero effects and how large those effects tend to be. The resulting posterior mean effects, rather than the original marginal GWAS estimates, are used as score weights.

Definition: Let $M$ be the number of SNPs in the modeled set, $N$ the GWAS sample size, $R$ the $M \times M$ LD correlation matrix, and $\hat{\boldsymbol{\beta}}$ the vector of marginal GWAS effect estimates. Let $\boldsymbol{\beta}$ denote the vector of underlying effects on the same scale. In LDpred’s standard approximation, with standardized genotypes and phenotype, $$ \hat{\boldsymbol{\beta}} \mid \boldsymbol{\beta},R \ \approx\ N\left(R\boldsymbol{\beta},\frac{1}{N}R\right). $$ The diagonal elements of $R$ are 1; its off-diagonal elements describe correlations between SNP genotypes. Thus, the expected marginal effect at a SNP can include contributions from effects at correlated SNPs.

This summary-statistic model is an approximation. It assumes the effect estimates come from a GWAS with appropriately scaled genotypes and phenotype, and that the LD matrix represents the correlations relevant to that GWAS. It also treats the reported summary statistics as having the covariance implied by $R/N$. These assumptions make it possible to work with summary statistics rather than individual-level GWAS genotypes.

The Sparse Causal-Effect Prior

The finite, or sparse, LDpred model assigns each SNP a probability $p$ of having a nonzero effect. Conditional on being causal under this model, an effect is drawn from a zero-mean normal distribution. A noncausal SNP has effect exactly zero. The prior variance is scaled so the expected total effect variance across the modeled SNPs is $h_g^2$, the SNP heritability on the standardized scale.

Definition: In the sparse LDpred prior, the effect at SNP $j$ follows the mixture $$ \beta_j \sim (1-p)\delta_0 + p\,N\left(0,\frac{h_g^2}{Mp}\right), $$ where $\delta_0$ is a point mass at zero. The expected number of causal SNPs is $Mp$, and the expected sum of prior effect variances is $Mp \cdot h_g^2/(Mp)=h_g^2$.

The causal indicators are usually assumed independent across SNPs before observing the GWAS data. The effects are centered at zero, so the model does not favor either effect direction in advance. After observing the data, however, the posterior distribution of one SNP’s effect depends on summary statistics at correlated SNPs through $R$. Consequently, a marginal estimate at a SNP is not generally enough to determine its posterior effect.

Worked Example: A SNP’s Posterior Inclusion Probability

Consider a simplified setting in which this SNP is independent of all others, so its marginal estimate has standard error $1/\sqrt{N}$. Suppose $N=1{,}000$, $M=100$, $h_g^2=0.20$, $p=0.10$, and the marginal estimate is $\hat{\beta}=0.050$. The prior variance conditional on a nonzero effect is $$ \tau^2=\frac{h_g^2}{Mp}=\frac{0.20}{100(0.10)}=0.020. $$

Under the null component, $\hat{\beta}\sim N(0,1/N)$; under the causal component, integrating over the normal effect prior gives $\hat{\beta}\sim N(0,\tau^2+1/N)$. The Bayes factor comparing the causal component with the null component is therefore $$ BF=\frac{1}{\sqrt{1+N\tau^2}}\exp\left\{\frac{N^2\tau^2\hat{\beta}^2}{2(1+N\tau^2)}\right\}. $$

Substitution gives $N\tau^2=20$, and $$ BF=\frac{1}{\sqrt{21}}\exp\left\{\frac{1{,}000^2(0.020)(0.050)^2}{2(21)}\right\} =\frac{1}{\sqrt{21}}\exp(1.1905)\approx 0.718. $$ The prior odds of a causal effect are $0.10/0.90=0.1111$. Thus the posterior odds are $0.1111(0.718)=0.0798$, and the posterior inclusion probability is $0.0798/(1+0.0798)\approx 0.0739$. Despite the positive estimate, this simplified model assigns about a 7.39% posterior probability to the causal component.

For this independent-SNP calculation, conditional on the causal component, normal-normal updating gives a posterior mean equal to the estimate multiplied by $N\tau^2/(1+N\tau^2)$. Here that conditional mean is $20/21 \times 0.050=0.0476$. The overall posterior mean also accounts for the possibility of a zero effect: it is approximately $0.0739 \times 0.0476=0.00352$. This illustrates why a posterior mean can be much closer to zero than the estimate conditional on causality.

With correlated SNPs, the same principle applies, but the posterior inclusion probability and effect at one SNP depend on the other SNPs’ summary statistics and their correlations. The one-SNP calculation is not a substitute for that joint analysis.

LDpred’s Joint Posterior and Posterior Mean Weights

Combining the summary-statistic likelihood with the sparse prior gives a joint posterior distribution for all effects. In the likelihood, the quadratic term involving $R$ couples effects at correlated SNPs. The posterior mean for SNP $j$ is the average of $\beta_j$ over this posterior distribution; it reflects both the possibility of a zero effect and uncertainty about the effect’s size.

For a finite $p$, the mixture prior is not a single multivariate normal distribution. LDpred’s original approach uses Gibbs sampling to approximate the joint posterior and its mean: it repeatedly draws each effect conditional on the current values of the other effects and the observed summary statistics. These posterior means are then used as weights. The procedure needs the GWAS summary statistics, an LD matrix, and choices or estimates for $p$ and $h_g^2$.

Worked Example: LDpred-inf With Two Correlated SNPs

LDpred-inf is the infinitesimal Gaussian-prior version of the model: it sets $p=1$, so all modeled SNP effects are assigned a Gaussian prior rather than a point-mass mixture. For standardized genotypes and phenotype, take $M=100$, $N=1{,}000$, $h_g^2=0.20$, and $R=\begin{pmatrix}1&0.5\\0.5&1\end{pmatrix}$ for a two-SNP illustration. This matrix is a valid correlation matrix because its eigenvalues, $1.5$ and $0.5$, are positive. Suppose the marginal estimates are $\hat{\boldsymbol{\beta}}=(0.080,0.060)^\mathsf{T}$.

Under the infinitesimal prior, $\boldsymbol{\beta}\sim N(0,(h_g^2/M)I)$. Combining this Gaussian prior with the summary-statistic likelihood gives the posterior mean $$ E(\boldsymbol{\beta}\mid\hat{\boldsymbol{\beta}},R) =\left(NR+\frac{M}{h_g^2}I\right)^{-1}N\hat{\boldsymbol{\beta}}. $$ Here, $M/h_g^2=500$, so the matrix on the left is $\begin{pmatrix}1500&500\\500&1500\end{pmatrix}$, and the right-hand vector is $(80,60)^\mathsf{T}$. Solving, $$ \begin{pmatrix}1500&500\\500&1500\end{pmatrix} \begin{pmatrix}\beta_1\\\beta_2\end{pmatrix} = \begin{pmatrix}80\\60\end{pmatrix} \quad\Longrightarrow\quad \begin{pmatrix}\beta_1\\\beta_2\end{pmatrix} = \begin{pmatrix}0.045\\0.025\end{pmatrix}. $$

The two posterior means are not simply the marginal estimates multiplied by one common shrinkage factor. The off-diagonal $0.5$ in $R$ means that each estimate contains information related to the other SNP’s effect. The posterior means account for that correlation as well as shrinking effects toward zero.

Why the Infinitesimal Posterior Formula Holds

The closed-form LDpred-inf result follows from completing the square in the product of the Gaussian likelihood and Gaussian prior. This also makes clear why the LD matrix appears in the effect estimates.

Derivation: Up to terms independent of $\boldsymbol{\beta}$, the log likelihood from the summary-statistic model is $$ -\frac{N}{2}\left(\boldsymbol{\beta}^{\mathsf T}R\boldsymbol{\beta} -2\boldsymbol{\beta}^{\mathsf T}\hat{\boldsymbol{\beta}}\right), $$ and the log prior under $p=1$ is $$ -\frac{M}{2h_g^2}\boldsymbol{\beta}^{\mathsf T}\boldsymbol{\beta}. $$ Adding them gives a quadratic with precision matrix $A=NR+(M/h_g^2)I$ and linear term $N\hat{\boldsymbol{\beta}}$. Completing the square, $$ -\frac{1}{2}\left(\boldsymbol{\beta}-A^{-1}N\hat{\boldsymbol{\beta}}\right)^{\mathsf T} A\left(\boldsymbol{\beta}-A^{-1}N\hat{\boldsymbol{\beta}}\right) +\text{constant}. $$ The posterior is therefore normal with mean $A^{-1}N\hat{\boldsymbol{\beta}}$, as stated. The added prior precision $(M/h_g^2)I$ regularizes the estimate, while $R$ accounts for dependence among SNPs.

This derivation assumes the specified Gaussian prior and summary-statistic likelihood. It does not establish the sparse model’s posterior means: when $p<1$, the point mass at zero makes the prior a mixture, so the single Gaussian completion-of-the-square calculation no longer gives the full posterior.

From Posterior Effects to a Polygenic Score

After estimating posterior mean effects, LDpred constructs a score by multiplying each person’s genotype by the corresponding weight and summing across the modeled SNPs. Effect-allele harmonization and consistent genotype scaling are essential: each genotype must count the allele to which its weight refers, and the genotype scale must match the effect scale used in estimation.

Definition: For an individual with standardized genotype vector $\mathbf{z}$, the LDpred score is $$ S_{\mathrm{LDpred}}=\sum_{j=1}^{M}z_j\,E(\beta_j\mid\hat{\boldsymbol{\beta}},R). $$ If the phenotype and genotypes were standardized, the score is on the model’s standardized phenotype scale. With differently scaled genotypes or phenotype, its units follow those scales.

Worked Example: Calculating a Score From Posterior Mean Effects

Use the two posterior means from the preceding example, $0.045$ and $0.025$. Suppose a hypothetical individual has standardized genotype values $z_1=1.0$ and $z_2=-1.0$ at the two SNPs. The score is $$ S_{\mathrm{LDpred}}=(1.0)(0.045)+(-1.0)(0.025)=0.045-0.025=0.020. $$

Thus this individual’s score is $0.020$ standardized phenotype units under the stated scaling. It is a weighted genetic predictor, not a guarantee of an outcome or evidence that either SNP causes a phenotype change in this individual.

Assumptions, Interpretation, and a Common Pitfall

LDpred’s estimates depend on the model, not only on the observed GWAS statistics. The sparse model assumes a specified causal fraction $p$ and a particular distribution of nonzero effects; LDpred-inf instead assumes a Gaussian effect distribution across all modeled SNPs. Both rely on the summary-statistic likelihood and an LD matrix appropriate to the genetic correlations represented in the GWAS. Posterior mean weights are therefore conditional on these modeling choices and inputs.

A common mistake is to treat the marginal GWAS estimate as the effect to assign to a SNP and then use LDpred only as a second, unrelated shrinkage step. That misses the joint model: the posterior effect depends on the pattern of marginal estimates and the LD correlations among the SNPs. Another mistake is to interpret an LDpred posterior mean as proof that a SNP is causal. A posterior mean combines uncertainty about whether an effect is zero with uncertainty about its size; it is a model-based weight, not a causal verdict.

In practice, the score is useful because it replaces a collection of noisy, correlated marginal estimates with weights that incorporate both prior shrinkage and LD. Its predictive performance still needs assessment in the intended target population; constructing posterior weights alone does not establish how accurately the score predicts outcomes there.

Check Your Understanding

Answer each question using the model definitions and calculations in this tutorial.

In the sparse prior, what does $p$ represent, and why is the conditional effect variance $h_g^2/(Mp)$?
Why can a SNP’s posterior effect depend on summary statistics from other SNPs?
How does the LDpred-inf prior differ from the finite sparse prior?
For standardized genotypes, how is an individual’s LDpred score calculated from posterior mean effects?
Why is it incorrect to interpret a nonzero posterior mean weight as proof that its SNP is causal?